The Symmetric Hyperbolic Distribution

III. The Symmetric Hyperbolic Distribution

In 1977, Barndorff–Nielsen 1977 introduced the family of generalized hyperbolic functions. The d-dimensional density function H d has six parameters: d and D for multivariate scales, m for location, b for skewness, and a and l mainly change the tails. 8 We have H d ~a, b, d, m, D, l~ x 5 k l a ~1 2d2l ~2p ~1 2d d l K l ~dk K l2~ 1 2d ~ad 2 1 ~ x 2 mD 21 ~ x 2 m T 1 2 d 2 1 ~ x 2 mD 21 ~ x 2 m T 1 2~~1 2d2l e b~ x2m T , 3 with k 5 =a 2 2 bDb T , D a symmetric, positive definite d 3 d-matrix with determinant 1 and K n the modified Bessel function of the third kind. 9 It is a well known theorem of Blæsild, that the family of generalized hyperbolic distributions is closed under regular affine transformations and with respect to formation of marginal and conditional distri- butions. 10 Blæsild explicitly describes the parameters of marginal, conditional or affine transformed hyperbolic distributions. A direct result of this theorem are the following lemma 2 and 3: 11 Lemma 2: The parameter b is 0, iff the corresponding parameter bˆ of any one dimensional marginal distribution is 0, i.e. the generalized hyperbolic distribution is elliptic, iff all one dimensional marginal distributions are symmetric. The one dimensional marginal distributions are normal hyperbolic distributions, iff the generalizing parameter l is 1. A generalized hyperbolic distribution with parameters b 5 0 and l 5 1 is therefore called a symmetric hyperbolic distribution. We denote the density E and replace the parameter a with z 5 def ad : E d ~z, d, m, D~x 5 z d 1 ~2p ~1 2d d d K 1 ~z K 12~1 2d S z Î 1 1 x 2 m d D 21 x 2 m T d D S z Î 1 1 x 2 m d D 21 x 2 m T d D ~1 2d21 . 4 The scale matrix of E is d 2 D. We can specialise Blæsild’s theorem to the case of symmetric hyperbolic distributions. Lemma 3 then states that the subfamily of symmetric hyperbolic distributions is closed with respect to forming marginal distributions and regular affine transformations. Lemma 3: Let X be a d-dimensional random vector distributed according to the symmetric hyperbolic distribution E d z, d, m, DX. Given a partition X 1 , X 2 of X, with 8 a , d and l are real, m and b vector parameters. Using l 5 1 ⁄ 2 d 1 1 one gets the usual multivariate hyperbolic distribution with density k 1 2~d11 ~2p 1 2~d21 2ad 1 2~d11 K 1 2~d11 ~dk e a Î d 2 1 ~ x2mD 21 ~ x2m T 1b~ x2m T 9 Abramowitz and Stegun 1972 provide a description of the properties, approximation and computation algorithms of the Bessel functions. The modified Bessel functions of the third kind are also known as MacDonald functions. 10 See Blæsild 1981. 11 See Bauer 1998 for an explicit proof of the theorem and the corollaries with the used notation. 458 C. Bauer r the dimension of X 1 , and analogously defined partitions b 1 , b 2 and m 1 , m 2 . Let D have block representation D 5 S D 11 D 12 D 21 D 22 D , so that D 11 is a r 3 r-matrix. Then the following holds: 1. The distribution of X 1 is a r-dimensional symmetric hyperbolic distribution E r zˆ, dˆ, m ˆ , D ˆ with parameters zˆ 5 z, dˆ 5 d uD 11 u 12r , m ˆ 5 m 1 and D ˆ 5 uD 11 u 1r D 11 . 2. Let Y 5 XA 1 B be a regular affine transformation of X and iAi denote the absolute value of the determinant of A. Then Y is distributed according to the symmetric hyperbolic distribution E d z, d, m , D with z 5 z, d 5 diAi 1d , m 5 mA 1 B and D 5 iAi 2d A T DA. We then need the moments of the distribution. The generalized hyperbolic distribution is an exponential family with respect to b. So its moments can be derived from the partial derivatives of the logarithm of the densities norming constant. 12 We get E~ X 2 m 5 0, 5 E~~ X 2 E~X 2 5 Cov~X 5 K 2 ~z zK 1 ~z d 2 D, 6 E~~ X 2 E~X 3 i , j,k 5 and 7 E~~ X 2 E~X 4 i , j,k,l 5 K 3 ~z z 2 K 1 ~z d 4 ~D i ,l D j ,k 1 D j ,l D i ,k 1 D k ,l D j ,i . 8 The interpretation of the 4 parameters is: m is a parameter of location, D defines only the shape of the covariance matrix and d varies only its size, while z is independent of transformations of the parameters of scale and location and determines the relation between the tails and the waist of the distribution. However, a change in z will result in a change of the size of the covariance matrix. There also exists an efficient generator for generalized hyperbolic random variables. Atkinson 1982 suggested a two or three envelope rejection algorithm. It uses the fact that the generalised hyperbolic distributions are mixed distributions. If s 2 is an random variable from an generalized inverse Gaussian distribution with parameters l, d 2 and k 2 , F the Cholesky decomposition of D and Y a d-dimensional standard normal random vector, then X 5 sYF T 1 m 1 s 2 bD is distributed with respect to H d a, b, d, m, D, l. The algorithm for creating random variables from an generalized inverse Gaussian distribution with parameters l, d 2 and k 2 simplifies, if we restrict the parameters to l 5 1 and b 5 0. 13 Using small values of z, as those estimated from the data used, it takes an 12 The i th derivative of 2ln k l a 12d2l 2p 12d d l K l dk with respect to b is the i th cumulant of H d a, b, d , m, D, l. With some calculation we get E~ X 2 m 5 ­ ­ b ln S ~2p ~1 2d d l K l ~dk k l a ~1 2d2l D 5 dK l1 1 ~dk kK l ~dk bD , which simplifies to 0 for b 5 0. The higher order moments can be derived in the same way. 13 A description is given in the Appendix. Hyperbolic Value-at-Risk 459 ISP-macro 14 less than 1 min to generate 10 000 generalized inverse Gaussian random variables. The efficiency of the algorithm relies only on z and the algorithm creating normal and uniformly distributed random variables. Creating high dimensional hyperbolically distrib- uted random variables and a Monte–Carlo evaluation of a portfolio with many assets one faces no more problems than if normal random vectors were used.

IV. Data and Estimation: A Comparison of the Gaussian and the Hyperbolic Model